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Welcome to ECE 329 Electrostatics – static electricity was know as an interesting but mystifying phenomenon of no practical use. ● Lightening ● Amber when rubbed attracted light objects to itself. ● Electrified fish and eels. History of Electromagnetics Magnetostatics – the existence of magnetic fields was known only through the behavior of naturally magnetized stones – lodestones. Certainly the Chinese, and perhaps the Central Americans figured out how to utilize the properties to make lodestone. the Reader's Digest version Light – of all the phenomena, light and its behavior were probably the most understood – lenses have been found that suggest that in the very early years AD the concepts of reflection and refraction were understood and exploited Sir Isaac Newton antiquity Some time passes 1643-1727 1770 Tranatlantic cable Maxwell 1866 1831-1879 State-of-the-art: 1770. There are two types of electricity (or maybe one that can be added to some materials and taken away from others) ● Electricity is conserved. ● There are insulators in which charge cannot move. ● There are conductors in which it can move freely. ● Equal charges repel each other. ● Opposite charges attract. ● Light reflection and refraction were understood and Sir Isaac Newton had put forth the notion that light was made of very small particles. ● And so it begins... QUESTION: How do we even know that electricity exists – or anything else really? The answer to this question is at the heart of understanding electromagetics and all of physics really. Answer: We know about our world because we can observe the effects of the existence of – say other matter or charged particles. From direct experience we know that gravity exists because everything that we drop is always pulled downward. In the case of electric phenomena we know how charged particles affect each other through experimentation. The simple law that we will learn is one of 2 fundamental pieces of evidence that constitutes all of electromagnetics. INTERACTION between CHARGES Charges exert a force on one another according to these observation: magnitude of the force is proportional to the magnitude of both charges ● magnitude of the force is inversely proportional to the square of the r1 distance between them ● direction of the force lies along the direction of a line connecting the charges. Like charges repel, unlike attract. ● +Q MATH Concepts - +q force coordinate systems vector notation ● fields ● ● Mapping the influence of a charged particle: +Q r1 A test charge is put in the vicinity of the charge. By moving the charge and measuring the force at each point a map can be created. Test charge: +q F1 Mapping the influence of a charged particle: +q r2 +Q F2 r1 x 1, y1, z 1 F1 To completely map the influence of Q at a point several pieces of information are needed: i) Label to specify the point in 3D space that the measurement was made. ii) Label to specify where the charge resides. ii) method of specifying the strength of the force (length of arrow) Iii) method of specifying the direction of the force (direction arrow points) Mapping the influence of a charged particle: +q +Q r2 F2 direction of the force lies along the direction of a line connecting the charges. Like charges repel, unlike attract. ● r1 x 1, y1, z 1 Qq ∣F 1∣= 2 4 o r 1 magnitude of the force is proportional to the magnitude of both charges ● magnitude of the force is inversely proportional to the square of the distance between them ● Map of the force that a test charge feels in the vicinity of charge +Q +q r2 x y z F2 2, 2, 2 r1 x 1, y1, z 1 +Q Map of the force that a test charge feels in the vicinity of charge +Q FIELD LINES +q r2 x y z F2 2, 2, 2 r1 x 1, y1, z 1 +Q To description of how +Q influences the test charge q at all points in space mathematically we need to find suitable operators and a geometry that fits our problems. STEP 1: DEFINE A COORDINATE SYSTEM specify origin ● define method of specifying a point ● define a method of specifying a vector ● Hi can you come to visit me? I live at 40.3256 oN 88.1293 oW ???????? SURE! How do I get there? I have my dad's flying caflemal. ?????? Maybe you better come over here. I live on face 4, 234.1 +60 12.34 -60. COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● z 3 mutually orthogonal axes following the right-hand convention y x Rene Descarte invented what is now called the Cartesian Coordinate system, named in his honor. This achievement happened around 1637... Without this innovation, the complex mathematics needed to quantify physical concepts would be impossibly difficult. Consider trying to describe quantitatively, using only words and some hand drawn figure how light behaves when going through a lens, for example. COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● z isx a z R : position vector plane z=z o parallel to x-y plane, intersecting the z-axis at z o zo (xo, yo, zo) yo y x xo x a x is GENERAL x y axis COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● z isx a z R : position vector zo R =x o x y o y z o z z y y x x x a x is GENERAL yo y axis COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● z isx a z unit vectors ALWAYS parallel to the axes z F 1 z x F 2 x y F 1=F 1x x F 1y y F 1z z F 2=F 2x x F 2y y F 2z z y axis y x a x is GENERAL x y COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● +q r2 +Q x y z F2 2, 2, 2 r1 x 1, y1, z 1 As an example let's look at describing this simple containing a single charge. We want to specify – in a compact way – the force at each point in space. COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● y +q r2 +Q x y z F2 2, 2, 2 r1 x 1, y1, z 1 origin x COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● Point where charge is located can be specified by specifying the equations of three surfaces, defining the point as their intersection. y (xs, ys, zs) (x2, y2, z2) +Q F2 Rs r2 (x1, y1, z1) r1 F1 In vector notation special vectors are used to specify the location of single points – POSITION Vectors Rs= x s x y s y z s z r1 =x 1 x y1 y z 1 z r2 =x 2 x y 2 y z 2 z x origin COORDINATE SYSTEMS specify origin ● define method of specifying a point ● define a method of specifying a vector ● y z y +Q F2 x z y x F1 x origin FIELDS ELECTRIC FIELD – measure of the force that a unit charge would feel at any location if it were there -Q force r , t=q E r , t F +Q FIELDS ELECTRIC FIELD – measure of the force that a unit charge would feel at any location if it were there -Q force r , t=q F E r ,t +Q Maxwell equations in integral and differential form MATH Concepts ● vector products E – electric field B – magnetic field (well not exactly) the basis for these equations and all of the interesting phenomena are based on 2 very simple concepts: i) simple interaction between charged particles ii) delayed response to change LORENTZ FORCE Now that we have some notation and the concept of the definition of a field we can investigate how these 2 simple concepts lead to the equation set that is the basis for electromagnetic analysis. i) simple interaction between charged particles LORENTZ FORCE – if a charge particle moves in a space where electric and magnetic fields exist it will feel a force generated by both fields in accordance with the following formula: =q E q v × F B it is clear that, depending on one's frame of reference q, the charge particle experiencing the force may or may not be moving. Forces and charge are both quantities that are invariant under changes in reference frame. WHAT DOES THIS IMPLY ABOUT E and B? LORENTZ FORCE i) delayed response to change – this is the property that has allowed electromagnetic theory to remain in tact despite upheavals in the approach to physics brought on by the advent of relativity and quantum mechanics. What happens to the force that this charge feels? -Q from t=-∞ to to the green charge is stationary moves with velocity v for dt=5s -Q from t = to+v*dt to ∞ the green charge is again stationary REVIEW Problems Coordinate systems/vector notation Find the distance between two points in space. ● adding and subtracting vectors Find the amount of work needed to move charges in electric and magnetic fields. adding and subtracting vectors ● dot (or scalar, or inner) product ● Calculate the angular momentum of a rotating charge. ● adding and subtracting vectors ● cross (or vector, or outer) product REVIEW Problems Coordinate systems/vector notation Find the distance between two points in space. ● adding and subtracting vectors B A REVIEW Problems Coordinate systems/vector notation Find the distance between two points in space. ● adding and subtracting vectors y establish coordinate system and units A B x REVIEW Problems Coordinate systems/vector notation Find the distance between two points in space. ● adding and subtracting vectors y d= 2−6 3−1 2 (2,3) A d (6,1) B x 2 REVIEW Problems Coordinate systems/vector notation Find the distance between two points in space. ● adding and subtracting vectors RB− RA=6−2 x 1−3 y y RB RA=62 x 13 y y d =∣ RB−R A∣= RB −R A ⋅ RB −R A =6−221−32 RA=2 x 3 y RA=2 x 3 y A A d RB =6 x 1 y d RB =6 x 1 y B B x x REVIEW Problems Coordinate systems/vector notation Find the amount of work needed to move charges in electric and magnetic fields. ● adding and subtracting vectors ● dot (or scalar, or inner) product A (6,9) Find the work needed to move the charge from point A to point B =1 y V E m distance between A and B: d tilt angle: θ charge: q B (2,13) REVIEW Problems Coordinate systems/vector notation Find the amount of work needed to move charges in electric and magnetic fields. ● adding and subtracting vectors ● dot (or scalar, or inner) product F ║ =? F =? F A (6,9) =1 y V E m ∣d =work ∣F ║ B (2,13) REVIEW Problems Lorentz Force calculate the position and velocity of a particle in an electric and/or magnetic field calculate the electric and magnetic fields present by the motion of charged particles REVIEW Problems Lorentz Force calculate the position and velocity of a particle in an electric and/or magnetic field =m a =q E F ma x =qE x ma y =qE y ma z =qE z REVIEW Problems Lorentz Force calculate the position and velocity of a particle in an electric and/or magnetic field Suppose Ex=Eo and Ey=Eo What do we expect to happen? dv x ma x =m =qE o dt or dv x q = Eo dt m this is a simple differential equation that can be solved by simply integrating in time to get - q v x = E o tv o m REVIEW Problems Lorentz Force ? q v x = E o tv o m x= ? q E o t 2 v ot x o 2m The velocity is shown to increase linearly in time going faster and faster with no limit in the direction of the field – IS THIS CORRECT? REVIEW Problems Lorentz Force calculate the position and velocity of a particle in an electric and/or magnetic field =m a =q E F ma x =qE x ma y =qE y ma z =qE z REVIEW Problems Lorentz Force calculate the position and velocity of a particle in an electric and/or magnetic field Suppose Bz=Bo What do we expect to happen? dv x m =qB o v y dt dv y m =−qB o v x dt =q v × F B this is a simple differential equation set of second order with st constant coefficients whose solutions are of the form v x = Ae 2 d vx dt 2 2 2 q Bo m 2 v x =0 REVIEW Problems Lorentz Force calculate the electric and magnetic fields present by the motion of charged particles A constant electric field and a constant magnetic field MAY exist – either or both can be present. Suppose you want to know the magnitude and direction of the fields and all you have is a single test charge. What experiment would you perform? vx F1 move charge a little bit in the x-direction and measure the force vy F 2 move charge a little bit in the y-direction and measure the force vz F3 move charge a little bit in the z-direction and measure the force REVIEW Problems Lorentz Force F 1x =qE x F 1y =qE y −q v x B z F 1z =qE z q v x B y F 2x=qE x qv y B z F 2y =qE y F 2z =qE z −qv y B x F 3x =qE x −qv z B y F 3y =qE y qV z B x F 3z =qE z measured quantities known experiment parameters desired quantities vx F1 move charge a little bit in the x-direction and measure the force vy F 2 move charge a little bit in the y-direction and measure the force vz F3 move charge a little bit in the z-direction and measure the force REVIEW Problems Lorentz Force calculate the electric and magnetic fields present by the motion of charged particles A constant electric field and a constant magnetic field MAY exist – either or both can be present. Suppose you want to know the magnitude and direction of the fields and all you have is a single test charge. What experiment would you perform? F1=0 V x =0 F2=0 V y =0 F3= A z V z =1 no information about the magnetic field F1=0 V x =0 F2=0 V y =1 F3 = A z V z =1 F1=0 V x =0 F2=0 V y =0 F3= A z V z =1 some examples – charged particles moving in an antenna what we know as propagating electromagnetic radiation could be computed this way, but it is completely impossible analytically using these two fundamental properties. It is almost impossible to compute the field of a single moving charge this way, but with perseverience... It was Faraday's and Maxwell's genius – particularly Maxwell to realize that the fields themselves have some very simple properties at a more abstract level which are more amenable to analysis. The sources and fields can be simply connected and – the fields can be substituted for the sources in many cases – radiation is a good example.